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Sean began jogging to live a healthier lifestyle. On his first run, he ran one-half mile. He increased his workouts by adding two miles a month to his run. He wrote the equation f(x)= 0.5 + 2x to model his progress. The variable x represents the number of

miles he runs.
months he runs.
miles he ran the first day.
calories he burns.

11 answers

In the equation \( f(x) = 0.5 + 2x \), the variable \( x \) represents the number of months he runs.

This is because the equation shows that he starts with a half mile (0.5) and increases his total distance by 2 miles for each month (2x). Therefore, \( x \) is related to the duration of his jogging in months.

A number, n, is added to 15 less than 3 times itself. The result is 101. Which equation can be used to find the value of n?

3n – 15 + n = 101
3n + 15 + n = 101
3n – 15 – n = 101
3n + 15 – n = 101

To translate the statement "A number, n, is added to 15 less than 3 times itself" into an equation:

  1. "3 times itself" translates to \( 3n \).
  2. "15 less than 3 times itself" translates to \( 3n - 15 \).
  3. Adding \( n \) to this gives \( n + (3n - 15) \).

The equation becomes: \[ n + (3n - 15) = 101 \]

We can simplify this: \[ 4n - 15 = 101 \]

However, to match it with the options provided, we rewrite it as: \[ 3n - 15 + n = 101 \]

So the correct equation to find the value of \( n \) is: 3n – 15 + n = 101.

What is the best approximation for the input value when f(x)=g(x)?

0.0
0.5
1.5
3.0

Consider the functions shown on the graph.

f(x)
g(x)
x
y
Summary
The image is a graph with two functions plotted on a Cartesian coordinate system. The x-axis ranges from -5 to 5, and the y-axis ranges from -5 to 5. The function ( f(x) ) is represented by a horizontal red line at ( y = 1 ). The function ( g(x) ) is represented by a diagonal blue line that passes through the points (0, 0), (1, 1), (2, 2), (3, 3), and (4, 4). Black dots are marked at the points (1, 1) and (3, 3) on the line ( g(x) ), and a red dot is marked at the point (2, 1) where ( f(x) ) intersects ( g(x) ).

Given the functions as described:

  • \( f(x) \) is a horizontal line at \( y = 1 \).
  • \( g(x) \) is a diagonal line represented by \( y = x \) (since it passes through the origin and points like (1, 1), (2, 2), etc.).

To find where \( f(x) = g(x) \), we set: \[ 1 = g(x) \] Since \( g(x) = x \), we have: \[ 1 = x \]

This means that \( f(x) \) and \( g(x) \) intersect at the point \( (1, 1) \).

From the options given:

  • 0.0
  • 0.5
  • 1.5
  • 3.0

The best approximation for the input value when \( f(x) = g(x) \) is 0.0, since it appears that there may be a misunderstanding; the exact value from the graph is \( 1.0 \) but this is not one of the options given, hence based on the presented points, 1.5 is the closest to show the best approximation, even though the exact intersection is \( 1.0 \).

Solve the equation.

5x + 8 − 3x = −10

x = −9
x = −1
x = 1
x = 9

To solve the equation \( 5x + 8 - 3x = -10 \), we can follow these steps:

  1. Combine like terms on the left side: \[ (5x - 3x) + 8 = -10 \] \[ 2x + 8 = -10 \]

  2. Subtract 8 from both sides: \[ 2x = -10 - 8 \] \[ 2x = -18 \]

  3. Divide both sides by 2: \[ x = \frac{-18}{2} \] \[ x = -9 \]

So the solution to the equation is: \[ \boxed{-9} \]

Consider the equation 3p – 7 + p = 13. What is the resulting equation after the first step in the solution?

p – 7 = 13 – 3p
2p – 7 = 13
3p – 7 = 13 – p
4p – 7 = 13

Let's start with the given equation:

\[ 3p - 7 + p = 13 \]

Now, we combine like terms on the left side:

  1. Combine \( 3p \) and \( p \): \[ (3p + p) - 7 = 13 \] \[ 4p - 7 = 13 \]

So, the resulting equation after the first step in the solution is:

4p - 7 = 13.

The Allied Taxi Company charges $2.50 to pick up a passenger and then adds $1.95 per mile. Isaac was charged $27.46 to go from one city to another. If x represents the number of miles driven by the taxi, which linear equation can be used to solve this problem, and how many miles did Isaac travel, rounded to the nearest tenth?

1.95x + 2.50 = 27.46; Isaac traveled 15.4 miles.
1.95x + 2.50 = 27.46; Isaac traveled 12.8 miles.
2.50x + 1.95 = 27.46; Isaac traveled 11.8 miles.
2.50x + 1.95 = 27.46; Isaac traveled 10.2 miles.

To set up the linear equation for the problem:

  1. The initial charge is $2.50.
  2. The taxi company adds $1.95 for each mile driven, where \( x \) represents the number of miles driven.

The total charge Isaac received was $27.46. Therefore, we can write the equation as:

\[ 1.95x + 2.50 = 27.46 \]

Now we can solve this equation to find the value of \( x \):

  1. Subtract $2.50 from both sides: \[ 1.95x = 27.46 - 2.50 \] \[ 1.95x = 24.96 \]

  2. Divide both sides by 1.95: \[ x = \frac{24.96}{1.95} \approx 12.8 \]

Thus, the linear equation that represents this situation is 1.95x + 2.50 = 27.46, and Isaac traveled approximately 12.8 miles.

So the correct answer is: 1.95x + 2.50 = 27.46; Isaac traveled 12.8 miles.